Lattices

A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice.

Example. Determine whether the posets represented by each of the Hasse diagrams in Figure below are lattices.

Solution: The posets represented by the Hasse diagrams in (a) and (c) are both lattices because in each poset every pair of elements has both a least upper bound and a greatest lower bound. On the other hand, the poset with the Hasse diagram shown in (b) is not a lattice, since the elements b and c have no least upper bound. To see this note that each of the elements d, e and f is an upper bound, but none of these three elements precedes the other two with respect to the ordering of this poset.

Glossary

equivalence relation – отношение эквивалентности; partition – разбиение

partial order – частичный порядок; linear order – линейный порядок

well-ordered set – вполне упорядоченное множество; bound – граница

lattice – решетка